Gear teeth and method of designing the same



Feb. 10. 1925. S 1,525,642

A. B. COX

GEAR TEETH AND METHOD OF DESIGNING THE SAME Original Filed Sept. 28, 1922 1 6 Sheets-Sheet 1 S l7 ar/Y 2a 60 M0 M0 /50 e20 260 WITNESSES:

A. B. COX

GEAR TEETH AND METHOD OF DESIGNING THE SAME Feb. 10. 1925.

Original F'ilecl Sept. 8 2

6 Sheets-Sheet 2 INVENTOR RNE A 5 fax BY A ,1

ATTO

M20 ya.

WITNESSES:

Feb. 10. 1 25. 4 T 1,525,642

v I A. B. cox

G EAR TEETH AND METHOD OF DESIGNING THE SAME inal Filed Sept. 28, 1922 6 Sheets-Sheet 5 10 ac 30 0 54 ea 7a WITNESSES: INVENTOR flfiox.

ATTORN EY A. B. cox

GEAR TEETH AND METHOD OF DESIGNING THE SAME Feb. 10,1 25.

Original Filed Sept. 28, 1922 6.Sheets-Sheet 4 Feb. 10. 1925. 1,525,642

A. B. COX

GEAR TEETH AND METHOD OE DESIGNING THE SAME i inal med Sept. 28. 1922 s Sheets-Sheet 5 e Fif w n Q, R w w A WITNESSESz' a I INVENTQR llfiiox.

ATTORNEY and Methods of Designing the Same, of

Patented Feb. 10, 1925.

UNITED STATES ANTHONY 13. 001:, or PITTSBURGH, PENNSYLVANIA.-

GEAR TEETH AND METHOD OF DESIGNING THE p Application flied September 28, 1922, Serial No. 591,092; Renewed December 24, 51924.

' and State of Pennsylvania, have'invented a new and useful Improvement in Gear Teeth which. the following. is a specification.

-Myinvention relates to gear-Wheel mechanisms embodying either internal or external gear combinations.

Theprincipal object of my invention is to provlde a smooth running gear mechanism comprising a pair of meshing gears, wherein the vibrations incident to the operation of the usual type of gear-wheel mechanism are substantially eliminated.

Anotherobject of my invention is to provide-a air .ofmating gear wheels embodying esign, wherein the load upon each of thevteeth in' mesh is maintained substantially constant throughout the meshing period of the same.

Other objects of my invention will appear more fully hereinafter from the following descriptionof the nature, the mode of'operation and the constructional details. of the,

gear-wheel mechanism embodying my invention, i l

gear-wheel mechanism embodying a pair of meshing gear wheels, a non-integral number of pairs of teeth are in contact during-the operation thereof. That is to say, an even 3 and an odd number of pairs of teeth are suc :ce'ssi'vely in engagement throughout the period of operation of the gear-wheel mechanism'. S1nce,;in the ordinary spur gear, all the teeth of the meshing gear have the same width across the face'pf. the gear in adirection parallelto the gear axis, the total length offtcoth carrying the-load will be given by the}, roduct of the total number of pairs of teeti in'conta'ct and the width of the face of the gear, The width of the face being a constant quantity, the total length of tooth carrying the load will vary with the'number of'pairs of' teeth in contact, which as just pointed out; varies from instant to instant in an ordinary pair of gears.

,'When,aneven number of pairs of teeth are in contact, as'two pairs of teeth, the

power or load transmitted through the gearwheel mechanism is carried by the two pairs of teeth. When an odd number of pairs of vibration,

teeth are in contact, as three pairs of teeth, said power is carried by the three pairs of teeth. Hence, it can be seen that the individual teeth of the mechanism are subjected to rapidly varying loads which results in not only of the contacting teeth, but also of the driven gear wheel of the mechanism. The latter vibration is es ecially serious because it involves a vibratlon design may be obtained by any of the following methods, either singly or in combina' tion: First, by choosing a suitable number of teeth for gears of a given ratio; second, by choosing a suitable angle for the line of actionythird, by choosing a suitable relative length for the fddendum; and fourth,

- i v by increasing or decreasing the distance be I have observed that, m the customary.-

tween the centers of a pair of co-operating gears a suitable amount. I

The following methods are applicable to helical, worm and spiral bevel gears only; fifth,.by choosing a suitable helix angle for the-direction of the tooth across the face of the 'ear'gand sixth, by choosing a suitable widt for the face of the gear.

Myginvention further consists in the details of arrangement, operation and design hereinafterset forth and claimed and illustrated in, the accompanying drawing, wherein v Figure 1 st diagram showing the co-op crating teethT-bf a rack and pinion, together with certainbther details which will be used in the mathematical analysis of the first method of desi namely the choosing of a suitable nu er of teet ,for gears of a given ratio.

Fig. 2 is a curve diagram-:sliowing the re lation between the number of pairs of teeth in contact M,;" or M,,, as'ordinates, and the number, of teeth at in the inion as abscissze,

and is employed in the rst method of -design. 7

Fig. 3 is a curve diagram showing the relation between the total number of teeth M. in contact and the number of teeth n in the pinion for different gear ratios and for a standard Bi tooth.

Fig. 4 is a viewvsimilar to Fig. 3 for a 17 standard tooth.

Fig. 5 is a view similar to Fig. 3 but em ploying an addendum .of 0.182 P and a 20- involute. V

Fig. 6 is a View of a rack and pinion designed according to the third method of design.

Fig. 7 is a diagram showing a pair of meshlng gear Wheels, together with certain other details which will be used in the mathematical analysis of the fourth method of design. Fig. 8 is a view showing the fourth method of design applied to a pair of internal mating gear wheels;

Fig. 9 is a diagrammatic plan view of a pair of helical racks wherein the teeth are cut according to a right hand helix and according to a left hand helix, respectively.

FigJO is a detail longitudinal sectional view of the lower edges of the half of the helical rack having right hand teeth, together with a corresponding section of a helical pinion meshing therewith.

Fig. 11 is a detail longitudinal sectional view of the upper edges of the helical rack having left hand teeth, together with a corresponding section of a helical pinion meshing therewith.

The following defines the symbols hereinafter employed in the specification and drawing:

a" 431416. P :diametral or normal pitch in helical gears. =diametral pitch. :circular pitch. I X :circular pitch of helical gear in a plane perpendicular'to the axis of the gear. qS :angle of line of action or degree of I involute. a, :eifective angle of line of action in plane perpendicular to the axis of the helical gear. 7 y :efl'ective angle of line of action or degree of involute in spur gears with increased center distance. :number of teeth in pinion. :number of teeth in gear. a :efi'ective number of teeth in bevel or helical pinion.

N :efi'ective number of teeth in bevel or helical gear. R :pitch radius of gear. 7' :pitch radius of pinion.

- K. :fiective ratio of bevel or helical gears.

p P cos Lg l cos a M I cos (1;

M. =total number of pairs of teeth in contact in a pair of meshing gears. For the case of the gear and pinion M zM -l-M whereas for the case of the rack and pinion M,:M +M

A. :length of addendum.

q==ratio of length of addendum to circular pitch.

S :distance between the pitch circles of two meshing gears measured on the line of centers.

8" :amount of change of addendum of pinion teeth caused b S.

s" =amount of. change of addendum of gear teeth caused by S.

a: :helix angle of helical gear.

B of the front pitch cone angle of a.

bevel pinion.

W :width of gear necessary 6th method of design.

V :distance giving relative positions of the two halves ofhelical gears, as shown in Fig. 9.

to apply the The first method, namely, the choosing of a suitable number of teeth for gears of a given ratio, is based upon the following analysis:

Assuming that the teeth of a pair of meshing gear-wheels are so accurately cut that all the pairs of teeth that should theo-.

' retically be in contact are in contact, then the theoretical number of pairs of teeth in rack, n'= and we have M or M for the rack or gear ,lrespectively. That is to say, M,:(M or M +M The derivation of the formula for M for teeth of stand-ard'length is as follows: 30

contact may easily be calculated. 7 Moreover, the number of pairs of teeth in contact with gears of any ratio and any numbers of teeth may be calculated and plotted in curve form, all as hereinafter explained.

Referring toFigure 1, it is seen that the tooth-contacts are made only on the line of action and that the 'distance measured on the line of action from a point on one tooth equation of outside diameter of pinion.

to a corresponding point on the next. tooth is P cos 4:. In order to ascertain the total quation of line of action.

illlmbelf f Pairs of h t in Contact, Substituting this value of g in'the first it is necessary to find the total length LT of equation: L 1 the line of action and divide by P cos 1). I z q r The integral part of the number M indi- 1 fl tan ;f-'

cates the minimum number of pairs df teeth di that will always be in contact'during the operation of the gears, while the decimal part of the number indicates the relative length of time an additional pair of teeth will be in contact. In determining the total number of pairs of teeth M, in contact, it has been found more convenient to ascer- C It tain the number of pairs of teeth in con- Om Hung erms" tact M for the pinion only and then add to nt/v sec +2a2m tan 41' (n+1)=0. this the number of pairs of teeth in contact Solving:

M n fP cos l and Clearing fractions: v i i 21rM cos +n sin =;!;'n\/Sin +;3-,(n+1) Squaring: j 1

l-zr M cos +45rnM, sin cos +n sin =n sin +4n+ 4. Combining terms: I

. vr cos +vrnM sin cos -n1 0 I Substituting -M for M Solving for n: I

1 1r"M,, cos d 1rM, sin 5 cos -l The preceding formula holds for gears and ranks, as well as for pinions. For a Values of M, that is, M, or M may now be plotted against values of at using equa- 7c M, sin 4 cos '1=0 tion 2, as the form most convenient for plotting, and assuming a 14 Std. involute gear tooth, all as shown by curve a in Fig. 2.

In Fig. 3 shown a series of curves derived from curve a. of Fig. 2. illustrating the relation between the total number of teeth in contact M, as ordinates and the number of teeth in the pinion as abscissae, for different values of gear ratios. Curves b, 0, (Z and e; are for external gear wheel mechanisms having a 1 1 standard tooth and gear ratios of 1 to 1; 1 to 2; 1 to 4 and 1 to co that is, a rack and pinion. Curves f, g and h are for internal gear-Wheel mechanismshaving gear ratios of 1 to 1, 1 to 2, and 1 t0 1.

The dotted line curve 7- in Fig. 3 is plotted according to the formula which was developed in the October 14th, 1920 number of the American Machinist. The formula (4) indicates the limiting portions of the several curves in Fig. 3 beyond which tooth interference or imperfect action obtains.

responding to the number of teeth in the gear, thus obtaining the total number of 'pairs of teeth in contact M, for a particular gear ratio and for a pinion and gear having certain numbers of teeth. For example, for a 1 to 2'ra-tio of gears, and for a tooth pinion, M is 107 5, whereas M for the gear wheel which would of course have 100 teeth, 'is 1.175. The total number of-pairs of teeth in contact M fora gear-wheel having this particular gear-wheel ratio and number of teeth is, therefore, 2.250. This number, 2.250, indicates that there will always be two pairs of teeth in contact and that for approximately one-fourth of the time, a third pair of teeth will be in contact and sharing the load.

The formula for the number of pairs of teeth in contact for standard internal gears is found by a method exactly similar to the From an observation of the several curves of Fig. 3, it is seen that the standard 1 1 tooth does not lend itself particularly to the first method of designing gear-wheel mechanisms having the desired integralnumber of pairs of teeth in contact, except for certain gearv ratios lying between the l to 1 and 2 to 1 ratios and for pinions having from 26 to 35 teeth and internal gears with pinions having from 48 to 73 teeth and gear ratios lying between 2 to l and 1 to 1. The one-to-one ratio of internal gears of course has no meaning except as showing the theoretical limit of the number of pairs of teeth in contact as the gear ratio approaches unity.

Somewhat similar results were obtained by employing a gear-wheel mechanism having a standard length of tooth and a 22 line of action. However, as shown in the several curves of Fig. 4, wherein the total number of teeth incontact M, is plotted against the number of teeth in the pinion, a gear-wheel mechanism having a 17 involute or line of action and a standard length of tooth allows the use of any ratio of gears while still'maintaining two pairs of teeth continuously in contact. The foregoing Figures 2, 3 and 4 and the aboye formulae (1,2 and 5) illustrate the fact that the number of pairs of teeth in contact M for a given gearratio changes with variations in the angle of the line of action p,

thereby illustrating the second method of.

design.

As brought out in the foregoing figures, it is practically impossible to design a gearwheel mechanism,having a standard length of tooth, wherein only one pair of teeth is in contact throughout the operation of the mechanism. A gear-wheel mechanism having only one pair of teeth in contact, however, may be developed according to the following method.

The previously mentioned formula (4) for noninterference of gears was derived for teeth of standard length. If, now, the addendum be expressed as'a fraction q of the circular pitch P (1) for the number of teeth in contact becomes and the interference formula (4) takes the form rma-2n sin (6) Expressing the gear ratio -3 as k, the interference formula (6) becomes (2k+ 1 sin In Fig. 4, the dot-and-dash line is thelocus of n for different values of the'gear-ratiok, for standard gearswherein Solving ing the number of pairs of teeth in contact M for gear teeth of any length in external and internal gear mechanisms,respectively;

In the case of the rack, nof formula (8) becomes infinite and M substituted for M M sin b cos -g =0,

and v I sin cos 15, Combining (7) and (8) i sin oiM, sinqScos d -g Clearing: 2wM g sin cos 2m:g'-=.i

1r 9 sin owl-M sin di cos o'. Combining terms: M sin #1 cos +2M,',g sin cos Referring also to equation (9), we find that Solving:

M 2 -q1/ T Sin cos 4: sin cos (a m Q l q1/3+sin MT -M+Mp. SiI1 e 008 9 P 00S sin 4: cos 4: Solving: i mechanism having but a single pair of teeth sin (15 cos in contact at all times. v g'-=M (10) The fourth method of designin a pair of m intermeshing gear-wheels such t at an in-: When M =1, .tegral. number of pairs of teeth are consin cos stantly in mesh is that of increasing or del (l1) creasing the distance between the centers of Fon 20 stub-tooth gears, using formula (11) 9 Works out-to be approximately 0.1.82. The tooth contact curves showing' the relation between the total number of teeth in contact M and the number of pairs of teeth, in the pinion for different gear ratios are shown in Fig. 5, assuming an addendum of 0.182 P and a 20 involute. v Figure 6 shows a rack and pinion designed according to the above method and the appearance of the teeth.

Thus, it is seen that by employing the method just described,namely that of choosing a proper length for the tooth addendum, 7 which constitutes the third-method of design, it is possible to providea gear-Wheel the gears, said distance being calculated accordlng to the" following method. This method may, with suitable arrangements, be used while the gears are inoperation.

In Fig. 7 .is shown a pair of meshing gear wheels, the pitch circles of which are separated by an amount S. The effect of such separation is to changenot only the efl'ective line of action from an angle of obliquity to a diiferent angle r, but also to change the effective length of the addendum of the gear and pinion, the latter being altered by the amounts 8' and s", respectively. The value of 7 and of s and 8 may be calculated as follows:

Referring to Fig. 7 it is seen that the effective line of action is changed from an anon=R =R cos (12) and the line is drawn perpendicular to mn, also 0m=r =r cos 4: (13) and this line is drawn perpendicular to m-n. The right triangles (Om-O,,) and (OnO are, therefore, similar, since the sides of one are parallel to the sides of the other. The angles marked 7 are therefore equal. Moreover,

is therefore from If. S be expressed as a fraction 0) of r, then Cos 'y I r+ (26 8 (2 and (26) becomes 1' cos cos w? 7'+ l-l-m Equation (27) gives the angle of the effectiveline of action.

For the condition of rack and pinion,

cos 'y= w that is where 7c= m equation (27) reduces to cos 7 00s t. It will also be seen from Fig. 7 that the effect of increasing the distance between the centers of the gears an amount equal to S not only changes the obliquity of the line of action, but also, in effect, shortens the effective length of the addendumof the gear teeth an amount 8, and similarly, the effective length of the pinion teeth an amount equal to s. For the condition of one-to-one gear ratio, both gears, of course, have the same decrease in the effective length of the addendum.

In the design of internal gear-wheel mechanisms according to this method, the procedure is as follows:

In Fig.- 8 is shown a pair of internal mating gear wheels wherein the several portions thereof are identified by symbols corresp'onding in significance to those of Fig. 7 Triangles om0 and 0''n,-0 are similar since the sides of one are parallel to those of the other. The angles "marked 7 are therefore equal. Furthermore,

sec (30) Similarly, hzr sec zr cos sec y. (31) s'=r-h:r'r cos it secv'r. (32) The ratio of s to s" is, therefore, from equations (30) and (32),

Using the eifective length of addendum As' for an external inion, As" for an external gear, A+s' or a pinion mating with an internal gear and A-s" for an internal gear and ,also usin the effective angle of obliquity calculate by formula (27) or (39), thevvalues so obtained may be employed 1n the formulae (8) and (8) to obtain an integral number of pairs of teeth in contact in a given pair of gears. The foregoing illustrates the fourth method of design.

The following advantages obtain in a pair of gear wheels, the centers of which are separated a predetermined. amount S; namely, tooth interference is decreased, thereby allowing the use of a smaller number of teeth in the pinion than is ordinarily considered good practice and, in addition,.

the slip of the gear teeth is decreased, thereby resulting in a corresponding increase in the operating efficiency of the gear wheels.

The foregoing methods of design are applicable to any type of spur gear wheel mechanism embodying either internal or ex:

ternal gear combinations, also bevel gears or helical gears, and may be used either singly 1 in the. January 19, 1922 number of the or in combination to obtain the desired result. In applying the foregoing formulae (8 and 8 to bevel gears, however, the effective ratio and the effective number of teeth of the meshing gears must be used.

As shown in the American Machinist for January 19, 1922, the effective number of teeth N in a bevel gear pinion is equal to n, sec B (40) and similarly, the effective.

number of teeth N in a bevel gear is equal to N B (41). there shown tobe equal to K where K is the ratio of the number of teeth of the gear to the number of teeth of the pinion.

In applying formulae (8) and (8) to helical gear-wheel mechanisms, the effective ratio and the effective number .of teeth must also be used, according to formulae derived American Machinist. These formulae for only. ing a suitable helix angle for the direction The effective ratio K is determining the effective values differ from those for bevel gears as follows:

shafts V in z'n sec or (42) and N zN see a (42) n and K.=-K.. 3) I f The following methods of design, namely,

the fifth and sixth, apply to helical gears The fifth method is that of choosof the tooth across the pitch face of the gear wheel, while the sixth method is that of choosing a suitable width for the face of the gear wheel. Referring to the first method just mentioned, the design pro-- cedure is as follows:

1 In the American Machinist for January 19, 1922, it was shown that helical gears withparallelshafts could be dealt with in the same manner as ordinary spur gears using the effective value of the angle of line For helical vgear wheels having parallel of action in the plane of rotation, and expressing the relative length of addendum" in terms of the circular pitch in this plane. The equations for determining the effectlve values of the variables P q and for use in the formulae (8) and (8 are as follows:

P zP 'sec oz. .(44) Also from the definition of gAzgP gx= FQ cos a. (45) The eflective value of the angle of the line of action in this plane is found by the [tan 4; see a] (46). -Substituting the above determined values, formula (1) for the number of pairs of teeth in contact in this plane then becomes.

Since both a and g depend on the angle a for-their value, the value of M and therefore M maybe altered as desired by properly choosing the value of a. This illustrates the fifth method of spur gear design.

This type of gear may be considered as made up of a large number of spur gears of infinitesimally small width, and when the variables are sofchosen as to make M. an

I integer, each of these gears of infinitely small width may be considered as having a tooth coming into contact'just at the instant a tooth is leavingTcontac't. 'Thus, a

constantytotal tooth length may also be made to carry the load in this type of gear;

It is customary, i1 helical gears with parallel shafts to coml 1e on one shaft two If these gears show an integral number of pairs of teeth in contact, all as brought out in the following analysis. V

Fig. 9 is a diagrammatic plan view of a pair of helical racks l and 2, wherein the teeth are out according to a right hand helix and according to a left hand helix,.

' /having left hand teeth, as shown in 9, to ether with a corresponding sectionof a he ical pinion meshing therewith. Hence, it follows that the two halves 1 and 2 of the rack have been placed in such relation to each other that when a tooth of the lower half 2 is just beginning to make contact at A, a tooth of the upper half 1 is just finishing making contact at A. The width of either rack must be so chosen that when a tooth, say C, of the upper edge of the lower half 2 of the rack is just beginning to leave contact, a tooth, say C, of the upper edge of the upper half 1 of therack will have just finished leavingcontact. Hence, the same sum-total helical length of teeth will then be carrying the load transmitted by the gearing and a uniform steady driving effect will be obtained. A rack and pinion was chosen in order to make the foregoing explanation simple, but the same efi'ectsob- 'tain for any ratio of helical gears.

If it is desired to make the gear face wider than W each gear-half yvidth must be increased in increments of W.,, as indicated in broken lines in Fig. 9.

From Fig. 9, it is seen that when a gear half has a face width of 2W and when a tooth of the gear half 2 is just beginning to make contact at A, another tooth of the same gear half is just finishing making contact at A, and when a tooth, say C, is just beginning to leave contact, another tooth of the same gear half will have just finished leaving contact at C", so that the totall length of load-carrying tooth of the gear half is maintained substantially constant.

P cos 0:

which are fairly included in the language thereof, when read inconnection with the prior art, regardless of the details and functions set forth in the description or illustrated in the drawing.

I claim as my invention:

1-. A pair of co-operating gear wheels wherein a predetermined number of pairs of teeth are constantly in mesh.

2. A pair of co-operating gear wheels wherein a predetermined number of pairs of teeth on the driving gear wheel are constantly in mesh with a predetermined number of teeth on the driven gear wheel.

A gear-wheel mechanism comprising a pair of mating gear.wheels embodying a design wherein a constant integral number of pairs of teeth are in contact at all times.

4. A loaded gear-wheel mechanism comprising a pair of mating gear Wheels embodying a design wherein a constant total length of tooth is carrying the load at all times. I

5. A gear-wheel mechanism comprising a pair of mating gear Wheels embodying such design that the product of the total number of pairs of teeth in contact and the width of the face of either gear is a constant quantity.

6. A loaded helical gearing aggregate embodying such design that the total length of tooth carrying the load at every instant is substantially constant.

7. A. loaded helical-gearing aggregate embodying such design that every time any pair of teeth are beginning to make contact, another pair of teeth are at substantially the same time finishing making contact, and every time any pair of teeth are beginning to leave contact, another pair of teeth are at substantially the same time finishing leaving contact.

8. A loaded "ear-wheel mechanism coma u I b I prising a pair of mating gears having an nee 1,642 ,9

adjustable 'center-distance between them, the design being such asto make the total length of contacting teeth substantially independent of the rotative movement of the gears. 9. An involute gear-wheel mechanism comprising a pair of mating gears having an adjustable center-distance between them, the design being such as to make the total length of contacting teeth substantially independent of the rotative movement of the gears.

10. An involute gear-wheel mechanism comprising a pair of mating gear wheels embodying a design wherein a constant integral number of pairsofteeth are in contact at all times.

where n is the number of teeth in said gear, cp is the pressureangle', and q is the ratio of the addendum to the circular pitch.

13. A pair of meshing involute gears wherein the number of teeth between the line of centers and the outside diameter of one gear, plus the number of teeth between the line of centers and the outside diameter of the other gear is a substantially integral number, said number of teeth for either gear being where n is the number of teeth in said gear, q: is the pressure angle, 9 is the ratio 0 the addendum to the circular pitch, and the number of teeth on thesmaller of the gears being at least as large as the value given by the equation 21rq[k+ /li +(2k+l) sin (270+Z) sin 4: where k is the gear ratio.

14. A pair of meshing helical gears wherein the total face width W is given by the formula I I l sin a where P is the normal pitch and a is the helix angle.

15. A pair of double helicalgears wherein the face width W of each geafhalf is given by the formula a 2P sin a' where P is the normal pitch and a is the helix angle, the teeth of the two halves of each gear being staggered a distance v 21 cos a relatively to each other.

a In testimony whereof, I have hereunto subscribed my name this 27th day of September, 1922. r I v ANTHONY acox. i

and that the\ said Letters Patentshould b Certificate of Correction. It is hereby certified that in Letters Patent No. 1,525,642, gfianted February 10,

1925, upon the application of Anthony B. =CoX,,of Pittsburgh, Pennsylvania, for

an improvement in Gear Teeth and Methods of Designing the Same, errors appear in the printed specification requiring correction as follows: Page 2, fine 113, for the: character 41: read a; page line 112, for 471. 9 read 41: 1 in the' denominatorof equation .(6); page 5, line 40, for 1:9 sin qz read 1:9 sim? page 6, line 14,;for 41,, read 13,; pages 6 and 7, in equations (20) to (25), in elusive, (261), (27), (33), (33'), (34.) to (39), inclusive, and (47), and in the second e nation 1n claim ]3.'for the character Z each occurrence read 1 a e q 7 a P g 6,; line 90, for R read R page 7, line 34,- 'for N E read NescB; same a e, line 72,-for P read'P line 76, for 9' read (,1, A; same page, line 87 P g 7 for a read cp page 8, strike out the three equations between lines 54 and 55; e read with these corrections therein that the same may conform .to the record of the case in the Patent Oflice.

' Signed and sealed this 10th day of March, A. D. 192.5.

[SEAL] 1 I I KARL FENNING',

' Acting 'Oommissioner of Patents. 

